Animated Solution for Physics - Dual Nature of Matter and Radiation: For photo-electric effect with incident photon wavelength $\lambda$, the stopping potential is $V_0$. Identify the correct variation(s) of $V_0$ with $\lambda$ and $\frac{1}{\lambda }$.

Have you ever looked at a mathematical equation and wondered what it actually looks like in the real world? Physics is not just about crunching numbers; it's about translating the language of mathematics into visual stories. Today, we are going to dive deep into one of the most revolutionary equations in modern physics—Einstein's Photoelectric Equation—and see how it paints two completely different, yet equally beautiful, graphical pictures.

Imagine you are shining a light on a metal surface, and electrons are popping out. The energy of these electrons depends on the color, or wavelength, of the light you use. Our goal is to understand how the stopping potential (the voltage needed to stop these electrons) changes when we tweak the wavelength of the incident light. Let's break it down!

Everything starts with Einstein's brilliant insight. He proposed that light is made of tiny packets of energy called photons. The energy of a single photon is given by $\frac{hc}{\lambda }$. When this photon hits an electron, it must first pay a "toll tax" to the metal, known as the work function $W$. Whatever energy is left over becomes the maximum kinetic energy of the ejected electron.

Mathematically, we write this as:

Here, $V_0$ is the stopping potential, and $e$ is the charge of an electron. To see how $V_0$ behaves, let's isolate it by dividing the entire equation by $e$:

This is our master equation. It holds the secret to both of the graphs we need to analyze.

Let's look at our master equation through a different lens. What if we treat $\frac{1}{\lambda }$ as a single variable, say $x$? And let $V_0$ be our $y$.

Suddenly, the equation transforms into a very familiar shape:

This is the classic equation of a straight line! Let's identify the parts: - The slope $m$ is $\frac{hc}{e}$. Since $h$, $c$, and $e$ are all positive constants, our line will have a constant positive slope. - The y-intercept $c$ is $-\frac{W}{e}$. Because the work function $W$ is positive, the y-intercept is negative.

If we were to draw this, the line would start below the x-axis. However, in the physical world, a negative stopping potential means the photons don't have enough energy to eject electrons at all. Therefore, the physical graph only starts from the x-axis and goes up.

The point where it crosses the x-axis ($V_0=0$) is the threshold condition. At this point, $\frac{1}{\lambda }=\frac{1}{{\lambda }_0}$. This perfectly matches the straight-line graph shown in the options!

Now, let's ask a different question. What happens if we plot $V_0$ directly against $\lambda$, instead of $\frac{1}{\lambda }$?

Let's look at our master equation again:

This equation is of the form $y=\frac{A}{x}-B$. This is not a straight line. Because the variable $\lambda$ is in the denominator, $V_0$ and $\lambda$ have an inverse relationship. In mathematics, this creates a curve known as a rectangular hyperbola.

Let's visualize its behavior: - When $\lambda$ is very small, the term $\frac{hc}{e\lambda }$ becomes massive, meaning the stopping potential $V_0$ is very high. - As $\lambda$ increases, the value of $\frac{hc}{e\lambda }$ drops rapidly. The curve sweeps downwards, concave up. - Eventually, at a specific wavelength $\lambda ={\lambda }_0$ (the threshold wavelength), the stopping potential hits exactly zero. - For any wavelength longer than ${\lambda }_0$, the energy is too low to eject electrons, so the graph stops there.

This non-linear, decreasing curve perfectly matches the first graph in our options.

By simply changing what we plot on the x-axis—from $\lambda$ to $\frac{1}{\lambda }$—we transformed a curved hyperbola into a crisp, straight line. This is a powerful tool in physics! It allows us to verify experimental data easily. If an experiment yields a straight line when plotting $V_0$ against $\frac{1}{\lambda }$, we know Einstein's theory holds true.

So, the next time you see a graph in a physics problem, don't just guess the shape. Write down the governing equation, identify your $y$ and $x$ variables, and let the math draw the picture for you!